This is the abstract of the paper "Exponential decay of $2\times2$ operator matrix semigroups" by Andras Batkai and Klaus-Jochen Engel. For the whole paper send the command get evolve-l 99-00016 to: "listserv@rz.uni-karlsruhe.de". Instead you may get the paper at "http://ma1serv.mathematik.uni-karlsruhe.de/evolve-l/index.html" too. Abstract: In the first section we show the exponential decay of certain $2\times 2$ operator matrix semigroups in Hilbert spaces. In the second part we give sufficient conditions for the well-posedness of complete second order Cauchy problems and apply our results of Section~1 in order to study the asymptotic behavior of the solutions. Finally, in the last section we consider some typical applications for damped wave equations which illustrate our abstract results obtained in the previous sections.